Browsing by Subject "Inverse problem"

Ultrasonic guided lamb waves can be used to monitor structural conditions of pipes and other equipment in industry. An example is to detect accumulated precipitation on the surface of pipes in a non-destructive and non-invasive way. The propagation of Lamb waves in a pipe is influenced by the fouling on its surface, which makes the fouling detection possible. In addition, multiple helical propagation paths around pipe structure provides rich information that allows the spatial localization of the fouled area. Gaussian Processes (GP) are widely used tools for estimating unknown functions. In this thesis, we propose machine learning models for fouling detection and spatial localization of potential fouled pipes based on GPs. The research aims to develop a systematic machine learning approach for ultrasonic detection, interpret fouling observations from wave signals, as well as reconstruct fouling distribution maps from the observations. The lamb wave signals are generated in physics experiments. We developed a Gaussian Process Regression model as a detector, to determine whether each propagation path is going across the fouling or not, based on comparison with clean pipe. This binary classification can be regarded as one case of the different fouling observations. Latent variable Gaussian Process models are deployed to model the observations over the unknown fouling map. Then Hamiltonian Monte Carlo sampling is utilized to perform full Bayesian inference for the GP hyper-parameters. Thus, the fouling map can be reconstructed based on the estimated parameters. We investigate different latent variable GP models for different fouling observation cases. In this thesis, we present the first unsupervised machine learning methods for fouling detection and localization on the surface of pipe based on guided lamb waves. In these thesis we evaluate the performance of our methods with a collection of synthetic data. We also study the effect of noise on the localization accuracy.

In my mathematics master's thesis we dive into the wave equation and its inverse problem and try to solve it with neural networks we create in Python. There are different types of artificial neural networks. The basic structure is that there are several layers and each layer contains neurons. The input goes to all the neurons in the first layer, the neurons do calculations and send the output to all the neurons in the next layer. In this way, the input data goes through all the neurons and changes and the last layer outputs this changed data. In our code we use operator recurrent neural network. The biggest difference between the standard neural network and the operator recurrent neural network is, that instead of matrix-vector multiplications we use matrix-matrix multiplications in the neurons. We teach the neural networks for a certain number of times with training data and then we check how well they learned with test data. It is up to us how long and how far we teach the networks. Easy criterion would be when a neural network has learned the inversion completely, but it takes a lot of time and might never happen. So we settle for a situation when the error, the difference between the actual inverse and the inverse calculated by the neural network, is as small as we wanted. We start the coding by studying the matrix inversion. The idea is to teach the neural networks to do the inversion of a given 2-by-2 real valued matrix. First we deal with networks that don't have the activation function ReLU in their layers. We seek a learning rate, a small constant, that speeds up the learning of a neural network the most. After this we start comparing networks that don't have ReLU layers to networks that do have ReLU layers. The hypothesis is that ReLU assists neural networks to learn quicker. After this we study the one-dimensional wave equation and we calculate its general form of solution. The inverse problem of the wave equation is to recover wave speed c(x) when we have boundary terms. Inverse problems in general do not often have a unique solution, but in real life if we have measured data and some additional a priori information, it is possible to find a unique solution. In our case we do know that the inverse problem of the wave equation has a unique solution. When coding the inverse problem of the wave equation we use the same approach as with the matrix inversion. First we seek the best learning rate and then start to compare neural networks with and without ReLU layers. The hypothesis once again is that ReLU supports the learning of the neural networks. This turns out to be true and happens more clearly with wave equation than with matrix inversion. All the teaching was run on one computer. There is a chance to get even better results if a more powerful computer is used.